3.5.1 Cauchy with unknown location.
Consider estimating the location parameter of a Cauchy distribution with a known scale parameter. The density function is
be a random sample of size
the log-likelihood function of
based on the sample. – Show that
is the Fisher information of this sample. – Set the random seed as
and generate a random sample of size
. Implement a loglikelihood function and plot against
. – Find the MLE of
using the Newton–Raphson method with initial values on a grid starting from
. Summarize the results. – Improved the Newton–Raphson method by halving the steps if the likelihood is not improved. – Apply fixed-point iterations using
, with scaling choices of
and the same initial values as above. – First use Fisher scoring to find the MLE for
, then refine the estimate by running Newton-Raphson method. Try the same starting points as above. – Comment on the results from different methods (speed, stability, etc.).
3.5.2 Many local maxima
Consider the probability density function with parameter
A random sample from the distribution is
x <- c(3.91, 4.85, 2.28, 4.06, 3.70, 4.04, 5.46, 3.53, 2.28, 1.96, 2.53, 3.88, 2.22, 3.47, 4.82, 2.46, 2.99, 2.54, 0.52)
- Find the the log-likelihood function of
based on the sample and plot it between
- Find the method-of-moments estimator of
. That is, the estimator
is value of
is the sample mean. This means you have to first find the expression for
- Find the MLE for
using the Newton–Raphson method initial value
- What solutions do you find when you start at
- Repeat the above using 200 equally spaced starting values between
. Partition the values into sets of attraction. That is, divide the set of starting values into separate groups, with each group corresponding to a separate unique outcome of the optimization.
3.5.3 Modeling beetle data
The counts of a floor beetle at various time points (in days) are given in a dataset.
beetles <- data.frame( days = c(0, 8, 28, 41, 63, 69, 97, 117, 135, 154), beetles = c(2, 47, 192, 256, 768, 896, 1120, 896, 1184, 1024))
A simple model for population growth is the logistic model given by
is the population size,
is an unknown growth rate parameter, and
is an unknown parameter that represents the population carrying capacity of the environment. The solution to the differential equation is given by
denotes the population size at time
- Fit the population growth model to the beetles data using the Gauss-Newton approach, to minimize the sum of squared errors between model predictions and observed counts.
- Show the contour plot of the sum of squared errors.
- In many population modeling application, an assumption of lognormality is adopted. That is , we assume that
are independent and normally distributed with mean
. Find the maximum likelihood estimators of
using any suitable method of your choice. Estimate the variance your parameter estimates.
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